Power Reduction Formulas/Cosine to 4th/Proof 2
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Theorem
- $\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$
Proof
| \(\ds \cos ^4 x\) | \(=\) | \(\ds \paren {\frac {e^{i x} + e^{-i x} } 2}^4\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i x} + e^{-i x} }^4} {16}\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i x} }^4 + 4 \paren {e^{i x} }^3 \paren {e^{-i x} } + 6 \paren {e^{i x} }^2 \paren {e^{-i x} }^2 + 4 \paren {e^{i x} } \paren {e^{-i x} }^3 + \paren {e^{-i x} }^4} {16}\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{4 i x} + 4 e^{2 i x} + 6 + 4 e^{-2 i x} + e^{-4 i x} } {16}\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {3 + 4 \paren {\dfrac {e^{2 i x} + e^{-2 i x} } 2} + \paren {\dfrac {e^{4 i x} + e^{-4 i x} } 2} } 8\) | gathering terms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cos 2 x + \cos 4 x} 8\) | Euler's Cosine Identity |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $23 \ \text{(b)}$